TY  - JOUR
TI  - Parity of an odd dominating set
AB  - For a simple graph $G$ with vertex set $V(G)={v_1,...,v_n}$, we define the closed neighborhood set of a vertex $u$ as $N[u]={v in V(G) ; | ; v ; text{is adjacent to} ; u ; text{or} ; v=u }$ and the closed neighborhood matrix $N(G)$ as the matrix whose $i$th column is the characteristic vector of $N[v_i]$. We say a set $S$ is odd dominating if $N[u]cap S$ is odd for all $uin V(G)$. We prove that the parity of the cardinality of an odd dominating set of $G$ is equal to the parity of the rank of $G$, where rank of $G$ is defined as the dimension of the column space of $N(G)$. Using this result we prove several corollaries in one of which we obtain a general formula for the nullity of the join of graphs.
AU  - Batal, Ahmet
DO  - 10.31801/cfsuasmas.1051208
PY  - 2022
JO  - Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics
VL  - 71
IS  - 4
SN  - 1303-5991
SP  - 1023
EP  - 1028
DB  - TRDizin
UR  - http://search/yayin/detay/1146262
ER  -